On the ternary domain of a completely positive map on a Hilbert C*-module

We associate to an operator valued completely positive linear map $\varphi$ on a $C^{\ast }$-algebra $A$ and a Hilbert $C^{\ast }$-module $X$ over $A$ a subset $X_{\varphi }$ of $X,$ called '\textit{ternary domain}' of $\varphi$ on $X,$ which is a Hilbert $C^{\ast }$-module over the multiplicative domain of $\varphi$ and every $\varphi$-map (i.e., associated quaternary map with $\varphi$) acts on it as a ternary map. We also provide several characterizations for this set. The ternary domain \ of $\varphi$ on $A\$ is a closed two-sided $\ast$-ideal $T_{\varphi }$ of the multiplicative domain of $\varphi$. We show that $XT_{\varphi }=X_{\varphi }$ and give several characterizations of the set $X_{\varphi }.$ Furthermore, we establish some relationships between $X_{\varphi }$ and minimal Stinespring dilation triples associate to $\varphi$. Finally, we show that every operator valued completely positive linear map $\varphi$ on a $C^{\ast }$ -algebra $A$ induces a unique (in a some sense) completely positive linear map on the linking algebra of $X$ and we determine its multiplicative domain in terms of the multiplicative domain of $\varphi$ and the ternary domain of $\varphi$ on $X$.